Here are some tips which may help:

(1) First of all, if you have any security software installed (e.g. Avast, AVG, etc.), please make sure that the program itself and its database are both up to date.
If this does not help, then try completely disabling your security software (at your own risk) or make an exception for the JuiceboxBuilder executable file to see if this makes a difference.
(Older versions of Avast's Ransomware Shield have been known to interfere with the functionality of JuiceboxBuilder but this should not be a problem with up-to-date versions.)

(2) With JuiceboxBuilder-Pro closed, delete the following folder manually and then try running JuiceboxBuilder-Pro again:
Mac: /Users/Username/Library/Application Support/Adobe/AIR/ELS/JuiceboxBuilder-Pro
Windows: C:\Users\Username\AppData\Roaming\Adobe\AIR\ELS\JuiceboxBuilder-Pro

(3) With JuiceboxBuilder-Pro closed, please check to see if the following folder exists on your system:

C:\Users\Username\Documents\JuiceboxBuilder-Pro\.tempGallery

(I'm not sure what the equivalent location would be for a Mac but if you search for ".tempGallery" you should hopefully find it.)
If there is a .tempGallery folder in this location, please delete it manually.
The .tempGallery folder is created automatically by JuiceboxBuilder for internal use only (for the live preview window and as a temporary gallery folder until the gallery is saved on the 'Publish' tab).
It should automatically be deleted by JuiceboxBuilder-Pro on closing the application but, if this fails, it has been known to render the 'New Gallery...' and 'Open Gallery...' buttons inactive.

The only way that I've been able to replicate the symptoms that you describe myself is to save a gallery to a Google Drive folder (so I'd recommend not saving a gallery to a location which is being synced to an online file sharing service) and the problem seems to be due to a persistent .tempGallery folder (which has not been removed as expected).
You may not be using a Google Drive folder but a persistent .tempGallery folder might still be causing a similar problem.

If all else fails, try completely uninstalling both JuiceboxBuilder-Pro and Adobe AIR following the procedure below.

(1) Uninstall JuiceboxBuilder-Pro.
Mac: Delete the 'JuiceboxBuilder-Pro' file from the Applications folder and empty your Trash.
Windows: Use the uninstaller in the 'Control Panel -> Programs -> Uninstall a program...' list.

(2) After uninstalling JuiceboxBuilder-Pro, please check that there are no files are left behind (even from a failed installation) by manually deleting the following folders (if they exist) from your hard drive:
Mac:
/Applications/JuiceboxBuilder-Pro.app/
/Users/Username/Library/Application Support/Adobe/AIR/ELS/JuiceboxBuilder-Pro
/Users/Username/Library/Preferences/JuiceboxBuilder-Pro
Windows:
C:\Program Files (x86)\JuiceboxBuilder-Pro
C:\Users\Username\AppData\Roaming\Adobe\AIR\ELS\JuiceboxBuilder-Pro
C:\Users\Username\AppData\Roaming\JuiceboxBuilder-Pro
C:\Users\Username\Documents\JuiceboxBuilder-Pro
You may need to show hidden files to find some of the above locations in Windows File Explorer.

(3) Also, please follow the procedure above (Steps #1 - #2) for JuiceboxBuilder-Lite, too (if you have it installed).
JuiceboxBuilder-Lite and JuiceboxBuilder-Pro can both be installed and run side by side on the same computer but, being that you are experiencing problems, it might be wise to completely clear your system of all JuiceboxBuilder files before trying to reinstall JuiceboxBuilder-Pro.

(4) Uninstall Adobe AIR.

(5) Reinstall Adobe AIR.
You can download the latest version of Adobe AIR from here.

(6) Reinstall JuiceboxBuilder-Pro (and make sure that it is the latest version, v1.5.1) following the instructions here.

Hopefully this will help.
Please let me know how you get on.

Thank you for the update.
I'm glad you've found the cause of your problem and have managed to get JuiceboxBuilder-Pro up and running again (although it's certainly strange that Avast did not give you the opportunity to make an exception for JuiceboxBuilder-Pro the first time it encountered it).

I would be very grateful if you could provide me with the following information so that I can log a false positive report with the Avast team.

(1) The version of JuiceboxBuilder-Pro that you are using ('Help -> About JuiceboxBuilder-Pro...').

(2) The versions of Avast that you are using (both the program version and the database version).

(3) The element of Avast which triggers the warning (Ransomware Shield or something else).

Thank you.

Many thanks for the information. It is most appreciated.
I have now filed a false positive report with the Avast team (using their online form here) who will hopefully be able to clear JuiceboxBuilder-Pro's reputation in their database.